A high frequency (111.2-420 GHz) electron spin resonance study of the interlayer spin diffusion is presented in the conducting phases of the layered organic compounds, kappa-( BEDT-TTF)(2)Cu[N(CN)(2)]X (kappa-ET2-X), X = Cl or Br. The interlayer spin cross relaxation time T-x and the intrinsic spin relaxation time T-2 of single layers are measured as a function of temperature and pressure. Spin diffusion is two dimensional in the high temperature bad-metal phase (i.e., electrons are confined to a single molecular layer for longer than T-2). The interlayer electron hopping frequency nu(perpendicular to) = 1/(2T(x)) decreases along the bad-metal to Mott insulator crossover and increases along the bad-metal to normal metal (or superconductor) crossover. The density of states (DOS) is determined from a comparison of T-x and the interlayer resistivity. In the bad-metal phase it is four to five times larger than the DOS calculated from the electronic structure neglecting electron correlations. In kappa-ET2-X the DOS increases with pressure along the bad-metal to normal metal crossover. Results are compared with predictions of the dynamical mean field theory.